"Project Swift" sounds like the name of a military invasion or an Olympic marathoner's training plan. But it is actually the code name for a corporate buyout, inspired by a private-equity associate's fondness for singer Taylor Swift.

Labels like Project Token, the name Apollo Global Management LLC used to mask its purchase of children's restaurant favorite Chuck E. Cheese, or Project Fusion, the code for Kinder Morgan Inc.'s consolidation of its oil-and-gas holdings into a single company, are designed to keep...